Local-to-Global-rigidity of lattices in $SL_n(\mathbb{K})$

DOI10.5802/AIF.3490arXiv2008.07250WikidataQ114013445 ScholiaQ114013445MaRDI QIDQ6347215

Amandine Escalier

Publication date: 17 August 2020

Abstract: A vertex-transitive graph $mathcalG$ is called Local-to-Global rigid if there exists $R>0$ such that every other graph whose balls of radius $R$ are isometric to the balls of radius $R$ in $mathcalG$ is covered by $mathcalG$. An example of such a graph is given by the Bruhat-Tits building of $PS{L}_n(mathbbK)$ with $ngeq4$ and $mathbbK$ a non-Archimedean local field of characteristic zero.. In this paper we extend this rigidity property to a class of graphs quasi-isometric to the building including torsion-free lattices of $S{L}_n(mathbbK)$. The demonstration is the occasion to prove a result on the local structure of the building. We show that if we fix a $PS{L}_n(mathbbK)$-orbit in it, then a vertex is uniquely determined by the neighbouring vertices in this orbit.

Mathematics Subject Classification ID

Geometric group theory (20F65) Groups with a (BN)-pair; buildings (20E42)

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